Primality proof for n = 11549194661:
Take b = 2.
b^(n-1) mod n = 1.
9466553 is prime. b^((n-1)/9466553)-1 mod n = 2204506690, which is a unit, inverse 9124742838.
(9466553) divides n-1.
(9466553)^2 > n.
n is prime by Pocklington's theorem.